Spectral theory of extreme statistics in birth-death systems

Statistics of rare events, or large deviations, in chemical reactions and systems of birth-death type have attracted a great deal of interest in many areas of science including cell biochemistry, astrochemistry, epidemiology, population biology, \textit{etc.} Large deviations become of vital importance when discrete (non-continuum) nature of a population of ``particles'' (molecules, bacteria, cells, animals or even humans) and stochastic character of interactions can drive the population to extinction. I will briefly review the novel \textit{spectral method} [1-3] for calculating the extreme statistics of a broad class of birth-death processes and reactions involving a single species. The spectral method combines the probability generating function formalism with the Sturm-Liouville theory of linear differential operators. It involves a controlled perturbative treatment based on a natural large parameter of the problem: the average number of particles/individuals in a stationary or metastable state. For extinction (the first passage) problems the method yields accurate results for the extinction statistics and for the quasi-stationary probability distribution, including the tails, of metastable states. I will demonstrate the power of the method on the example of a branching and annihilation reaction, $A \to\hspace{-2.8mm}\hspace{2mm}2A\,,\,2A \to\hspace{-2.8mm}\hspace{2mm} \emptyset$, representative of a rather general class of processes. \begin{enumerate} \item{M. Assaf and B. Meerson, Phys. Rev. Lett. \textbf{97}, 200602 (2006).} \item{M. Assaf and B. Meerson, Phys. Rev. E \textbf{74}, 041115 (2006).} \item{M. Assaf and B. Meerson, Phys. Rev. E \textbf{75}, 031122 (2007).} \end{enumerate}